内蒙古师范大学数学科学学院，内蒙古 呼和浩特

收稿日期：2023年6月29日；录用日期：2023年7月31日；发布日期：2023年8月7日

摘要

本文首先引入不分明化拓扑空间中 $T_0^{S\theta },T_1^{S\theta },T_2^{S\theta },R_S^{\theta },N_1^{S\theta },R_0^{S\theta },R_1^{S\theta }$ 分离公理的定义，再利用Fuzzifying拓扑空间理论和连续值逻辑语义方法进行研究，得到不分明化 $\theta$ -半分离相关定理。

关键词

半分离公理，不分明化半 $\theta -R_0$ 分离性，不分明化拓扑空间

$\theta$ -Semiseparation Axioms in Fuzzifying Topology

Wenying Jia, Ruiying Wang

College of Mathematics Science, Inner Mongolia Normal University, Hohhot Inner Mongolia

Received: Jun. 29^th, 2023; accepted: Jul. 31^st, 2023; published: Aug. 7^th, 2023

ABSTRACT

We introduce the definitions of $T_0^{S\theta },T_1^{S\theta },T_2^{S\theta },R_S^{\theta },N_1^{S\theta },R_0^{S\theta },R_1^{S\theta }$ separation axioms in fuzzifying topology space, the fuzzy topological space theory and logical semantics of continuous values are used to prove main results, and fuzzifying $\theta$ -semiseparation axioms are obtained.

Keywords:Semiseparation Axioms, Fuzzifying Semi $\theta -R_0$ Separation Axioms, Fuzzifying Topology

Copyright © 2023 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

1968年C.L. Chang提出不分明拓扑空间的概念，此后不分明拓扑学得到了迅速的发展，而且对问题的分析讨论也在逐步深化，各个不同方向的研究都得出了一些比较深刻的结果。应明生教授 [1] [2] [3] 提出了不分明化拓扑的概念，并从不同的角度发展了不分明集框架下的拓扑学。1982年，Dorsett C提出一般拓扑空间中的半 $T_0,T_1,T_2,R_0,R_1$ 分离定理。1984年，胡庆平提出一般拓扑空间中 $S_3,S_4$ 分离定理。此后，张广济和F.H. Khedr提出Fuzzifying拓扑空间中的半 $T_0,T_1,T_2,T_3,T_4$ 分离定理和半 $R_0,R_1$ 分离定理。同时Alkazragy A和Caldas M在一般拓扑中提出 $\theta -semi{T}_0$ ， $\theta -semi{T}_1$ ， $\theta -semi{T}_2$ ， $\theta -semiR$ ， $\theta -semiN$ ， $\theta -semi{R}_0$ ， $\theta -semi{R}_1$ 分离定理，并展开相关研究。于是在前人基础上，如何将一般拓扑空间中 $\theta$ -半分离定理推广到不分明化拓扑空间中得到不分明化 $\theta$ -半分离定理，这对于丰富不分明化拓扑空间理论是重要的。

本文在前人工作的基础上在Fuzzifying拓扑空间中引入 $\theta$ -半分离定理，得到Fuzzifying $\theta$ -半分离定理的一些好的性质和结论。

2. 预备知识

定理1 [4] 设 $\left(X,T\right)$ 是不分明化拓扑空间, $\forall A\in P\left(X\right)$ ， $\models A\in N_x^{\theta }\to A\in N_x$ 。

定义1 [5] A的半闭包 $\underset{\_}{A}$ 定义为： $x\in \underset{\_}{A}:=\forall B\left(\left(B\supseteq A\right)\wedge \left(B\in {\mathcal{F}}_S\right)\to x\in B\right)$ 。

定义2 [5] 设 $\Omega$ 是Fuzzifying拓扑空间类，一元模糊谓词 $S_i\in F\left(\Omega \right),i=0,1,2$ 被称为是 $S_i$ 分离的，以下为一些等价定理：

$\begin{array}{l}\models S_0\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to \neg \left(x\in \underset{\_}{\left\{y\right\}}\right)\vee \neg \left(y\in \underset{\_}{\left\{x\right\}}\right)\right)\\ \models S_1\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S\right)\\ \models S_2\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to \exists B\left(B\in {\text{B}}_x^S\right)\wedge \left(y\notin \underset{\_}{B}\right)\right)\end{array}$

定义3 [6] A的 $\theta$ -半闭包定义为对所有满足 $x\in X$ 且 ${\underset{\_}{U}}^S\cap A\ne \varnothing$ 的集合，表示为 ${\underset{\_}{A}}^{S\theta }$ ， ${\underset{\_}{U}}^S$ 为U的半闭包。

定义4 [7] 当存在X的一个 $\theta$ -开集U满足 $U\subseteq A\subseteq Cl\left(U\right)$ ，则子集A称为 $\theta$ -半开集。其中 $Cl\left(U\right)$ 为U的闭包。

定义5 [8] 设 $\left(X,T\right)$ 是一个拓扑空间， $x\in X$ ， $M\subset X$ ，则M称为x的 $\theta$ -半邻域当存在一个包含x的 $\theta$ -半开集A满足 $x\in A\subset M$ 。

定义6 [9] 设 $\left(X,T\right)$ 是一个Fuzzifying拓扑空间，则一元模糊谓词 $T_S\in F\left(P\left(X\right)\right)$ 称为Fuzzifying半开集，若 $A\in T_S:=\left(\exists B\right)\left(\left(B\in T\right)\wedge \left(B\subseteq A\right)\wedge \left(\forall x\right)\left(x\in A\to x\in \bar{B}\right)\right)$ 。

定义7 [9] 设 $x\in X$ ， $N_x^S\in F\left(P\left(X\right)\right)$ 表示x的半邻域系，定义为：

$A\in N_x^S:=\exists B\left(\left(B\in T_S\right)\wedge \left(x\in B\subseteq A\right)\right)$

定理2 [10] (1) $A\subseteq \underset{\_}{A}$ ；(2) $A\equiv \underset{\_}{A}\leftrightarrow A\in {\mathcal{F}}_S$ ；(3) $\models \left(A\subseteq B\right)\to \left(A\in N_x^S\to B\in N_x^S\right)$ 。

3. 主要结果及其证明

首先，为了方便书写，下面给出一些简记记号：

$\begin{array}{l}S{K}_{x,y}^{\theta }:=\left(\exists A\right)\left(\left(A\in N_x^{\theta S}\right)\wedge \left(y\notin A\right)\vee \left(\left(A\in N_y^{\theta S}\right)\wedge \left(x\notin A\right)\right)\right)\\ S{H}_{x,y}^{\theta }=\left(\exists A\right)\left(\exists B\right)\left(\left(A\in N_x^{\theta S}\right)\wedge \left(y\notin A\right)\wedge \left(B\in N_y^{\theta S}\right)\wedge \left(x\notin B\right)\right)\\ S{M}_{x,y}^{\theta }:=\left(\exists A\right)\left(\exists B\right)\left(\left(A\in N_x^{\theta S}\right)\wedge \left(B\in N_y^{\theta S}\right)\wedge \left(A\cap B=\varnothing \right)\right)\end{array}$

定义1 设 $\left(X,T\right)$ 是一个Fuzzifying拓扑空间，则称一元不分明谓词 $T_S^{\theta }\in F\left(P\left(X\right)\right)$ 为Fuzzifying $\theta$ -半开集，若 $A\in T_S^{\theta }:=\left(\exists B\right)\left(\left(B\in T_{\theta }\right)\wedge \left(B\subseteq A\right)\wedge \left(\forall x\right)\left(x\in A\to x\in \bar{B}\right)\right)$ 。

定义2 设 $x\in X$ ， $N_x^{\theta S}\in F\left(P\left(X\right)\right)$ 表示x的Fuzzifying $\theta$ -半邻域系，定义为：

$A\in N_x^{\theta S}:=\exists B\left(\left(B\in N_x^S\right)\otimes \left(\underset{\_}{B}\subseteq A\right)\right)$

定理1 对 $\forall A\in P\left(X\right)$ ， $\models \left(A\in T_S^{\theta }\right)\leftrightarrow \left(\forall x\right)\left(x\in A\to A\in N_x^{\theta S}\right)$ 。

定义3 设 $\Omega$ 是Fuzzifying拓扑空间类，分别称一元模糊谓词 $T_i^{S\theta }\in F\left(\Omega \right),i=0,1,2$ 为是Fuzzifying $T_i^{S\theta }$ 分离的，定义为：

$\begin{array}{l}{T}_0^{S\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to S{K}_{x,y}^{\theta }\right)\\ T_1^{S\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to S{H}_{x,y}^{\theta }\right)\\ T_2^{S\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to S{M}_{x,y}^{\theta }\right)\end{array}$

定义4 $\forall A\in X$ ，Fuzzifying半 $\theta$ -闭包定义为：

$x\in C{l}_S^{\theta }\left(A\right):=\left(\forall B\right)\left(\left(B\in N_x^S\right)\to \neg \left(A\cap \underset{\_}{B}\equiv \varnothing \right)\right)$

定理2 $\models T_0^{S\theta }\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to \neg \left(x\in C{l}_S^{\theta }\left\{y\right\}\right)\vee \neg \left(y\in C{l}_S^{\theta }\left\{x\right\}\right)\right)$

证明

$\begin{array}{c}\left[T_0^{S\theta }\left(X,T\right)\right]=\underset{x\ne y}{\inf }\max \left(\underset{y\notin A}{\sup }{N}_x^{\theta S}\left(A\right),\underset{x\notin A}{\sup }{N}_y^{\theta S}\left(A\right)\right)\\ =\underset{x\ne y}{\inf }\max \left(N_x^{\theta S}\left(X~\left\{y\right\}\right),N_y^{\theta S}\left(Y~\left\{x\right\}\right)\right)\\ =\underset{x\ne y}{\inf }\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)\\ =\left[\left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to \neg \left(x\in C{l}_S^{\theta }\left\{y\right\}\right)\vee \neg \left(y\in C{l}_S^{\theta }\left\{x\right\}\right)\right)\right]\end{array}$

定理3 $\models x\in \underset{\_}{A}\to x\in C{l}_S^{\theta }(\; A\; )$

证明

$\begin{array}{c}\underset{\_}{A}\left(x\right)=\underset{B\in P\left(X\right)}{\inf }\min \left(1,1-N_x^S\left(B\right)+\underset{y\in A}{\sup }B\left(y\right)\right)\\ \le \underset{B\in P\left(X\right)}{\inf }\min \left(1,1-N_x^S\left(B\right)+\underset{y\in A}{\sup }\underset{\_}{B}\left(y\right)\right)=C{l}_S^{\theta }\left(A\right)(\; x\; )\end{array}$

定理4 $\models T_0^{S\theta }\left(X,T\right)\to S_0\left(X,T\right)$

证明 $\left[T_0^{S\theta }\left(X,T\right)\right]=\underset{x\ne y}{\inf }\max (\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)(\; y\; ))$

$\left[S_0\left(X,T\right)\right]=\underset{x\ne y}{\inf }\max \left(\left(1-\underset{\_}{\left\{y\right\}}\right)\left(x\right),\left(1-\underset{\_}{\left\{x\right\}}\right)\left(y\right)\right)$ ，则 $\models T_0^{S\theta }\left(X,T\right)\to S_0\left(X,T\right)$

定理5 $\models T_1^{S\theta }\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S^{\theta }\right)$

证明 对任意 $x_1\ne x_2$ ，

$\begin{array}{c}\left[\left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S^{\theta }\right)\right]=\underset{x\in X}{\inf }{T}_S^{\theta }\left(X~\left\{x\right\}\right)=\underset{x\in X}{\inf }\underset{y\in X~\left\{x\right\}}{\inf }{N}_y^{\theta S}\left(X~\left\{x\right\}\right)\\ \le \underset{x\in X~\left\{x_2\right\}}{\inf }{N}_y^{\theta S}\left(X~\left\{x_2\right\}\right)\le N_{x_1}^{\theta S}\left(X~\left\{x_2\right\}\right)\\ =\underset{x_2\notin A}{\sup }{N}_{x_1}^{\theta S}(\; A\; )\end{array}$

同理 $\left[\left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S^{\theta }\right)\right]\le N_{x_2}^{\theta S}\left(X~\left\{x_1\right\}\right)=\underset{x_1\notin B}{\sup }{N}_{x_2}^{\theta S}(\; B\; )$

则

$\begin{array}{c}\left[\left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S^{\theta }\right)\right]\le \underset{x_1\ne x_2}{\inf }\min \left(\underset{x_2\notin A}{\sup }{N}_{x_1}^{\theta S}\left(A\right),\underset{x_1\notin B}{\sup }{N}_{x_2}^{\theta S}\left(B\right)\right)\\ =\underset{x\ne y}{\inf }\min \left(\underset{y\notin A}{\sup }{N}_x^{\theta S}\left(A\right),\underset{x\notin B}{\sup }{N}_y^{\theta S}\left(B\right)\right)=\left[T_1^{S\theta }\left(X,T\right)\right]\end{array}$

反过来

$\begin{array}{c}\left[T_1^{S\theta }\left(X,T\right)\right]=\underset{x_1\ne x_2}{\inf }\min \left(\underset{x_2\notin A}{\sup }{N}_{x_1}^{\theta S}\left(A\right),\underset{x_1\notin B}{\sup }{N}_{x_2}^{\theta S}\left(B\right)\right)\\ =\underset{x_1\ne x_2}{\inf }\min \left(N_{x_1}^{\theta S}\left(X~\left\{x_2\right\}\right),N_{x_2}^{\theta S}\left(X~\left\{x_1\right\}\right)\right)\\ \le \underset{x_1\ne x_2}{\inf }\left(N_{x_1}^{\theta S}\left(X~\left\{x_2\right\}\right)\right)=\underset{x_2\in x_1}{\inf }\underset{x_1\in X~\left\{x_2\right\}}{\inf }\left(N_{x_1}^{\theta S}\left(X~\left\{x_2\right\}\right)\right)\\ =\underset{x_2\in X}{\inf }{T}_S^{\theta }\left(X~\left\{x_2\right\}\right)\end{array}$

所以， $\models T_1^{S\theta }\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\left\{x\right\}\in {\mathcal{F}}_S^{\theta }\right)$ 。

定义5 设 $\Sigma$ 是一类不分明化拓扑空间，一元模糊谓词 ${\mathcal{F}}_S^{\theta }\left(T_S^{\theta }\right)\in F\left(P\left(X\right)\right)$ 称为Fuzzifying $\theta$ -半闭集，定义为： $A\in {\mathcal{F}}_S^{\theta }:=A\equiv C{l}_S^{\theta }\left(A\right)$ ，即 ${\mathcal{F}}_S^{\theta }\left(A\right)={\displaystyle {\int }_{P\left(X\right)}\underset{x\in X\sim A}{\inf }\left(1-C{l}_S^{\theta }\left(A\right)\left(x\right)\right)/A}$ 。

定理6 $\models A\in {\mathcal{F}}_S^{\theta }\to A\in \mathcal{F}$ 。

证明 由定义5和定理3易证。

定理7 $\models T_1^{S\theta }\left(X,T\right)\to S_1\left(X,T\right)$

证明 $\left[T_1^{S\theta }\left(X,T\right)\right]=\underset{x}{\inf }{\mathcal{F}}_S^{\theta }\left(\left\{x\right\}\right)\le \underset{x}{\inf }{\mathcal{F}}_S\left(\left\{x\right\}\right)=\left[S_1\left(X,T\right)\right]$ 。

定理8 (1) $\models T_1^{S\theta }\left(X,T\right)\to T_0^{S\theta }\left(X,T\right)$ ；(2) $\models T_2^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)$

证明 (1) (2)由定义显然得证。

定义6 设 $\Omega$ 是不分明化拓扑空间类，一元模糊谓词 $R_S^{\theta }\in F\left(\Omega \right)$ 称为Fuzzifying半 $\theta -R$ 分离的，定义为：

$R_S^{\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall D\right)\left(\left(D\in {\mathcal{F}}_S^{\theta }\right)\wedge \left(x\notin D\right)\to \left(\exists A\right)\left(\left(A\in N_x^{\theta S}\right)\wedge \left(C{l}_S^{\theta }\left(A\right)\cap D=\varnothing \right)\right)\right)$

定理9 $\models R_S^{\theta }\left(X,T\right)\leftrightarrow \left(\forall x\right)\left(\forall A\right)\left(\left(A\in T_S^{\theta }\right)\wedge \left(x\in A\right)\to \exists B\left(\left(B\in N_x^{S\theta }\right)\wedge \left(C{l}_S^{\theta }\left(B\right)\subseteq A\right)\right)\right)$

证明

$\begin{array}{c}\left[R_S^{\theta }\left(X,T\right)\right]=\underset{x\notin D}{\inf }\min \left(1,1-{\mathcal{F}}_S^{\theta }\left(D\right)+\underset{B\in P\left(X\right)}{\sup }\min \left(N_x^{\theta S}\left(B\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(B^c\right)\right)\right)\\ =\underset{x\in A}{\inf }\min \left(1,1-T_S^{\theta }\left(A\right)+\underset{B\in P\left(X\right)}{\sup }\min \left(N_x^{\theta S}\left(B\right),\underset{y\in A^c}{\inf }{N}_y^{\theta S}\left(B^c\right)\right)\right)\\ =\left[\left(\forall x\right)\left(\forall A\right)\left(\left(A\in T_S^{\theta }\right)\wedge \left(x\in A\right)\to \exists B\left(\left(B\in N_x^{S\theta }\right)\wedge \left(C{l}_S^{\theta }\left(B\right)\subseteq A\right)\right)\right)\right]\end{array}$

定理10 $\models R_S^{\theta }\left(X,T\right)\underset{\dot{}}{\wedge }{T}_1^{S\theta }\left(X,T\right)\to T_2^{S\theta }\left(X,T\right)$

证明

$\begin{array}{l}\left[R_S^{\theta }\left(X,T\right)+T_1^{S\theta }\left(X,T\right)\right]\\ =\underset{x\notin A^c}{\inf }\min \left(1,1-T_S^{\theta }\left(A^c\right)+\underset{B\in P\left(X\right)}{\sup }\min \left(N_x^{\theta S}\left(B\right),\underset{y\in A^c}{\inf }{N}_y^{\theta S}\left(B^c\right)\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right)\\ \le \underset{x\in X,y\ne x}{\inf }\underset{y\in X}{\inf }\min \left(1,1-T_S^{\theta }\left({\left\{y\right\}}^c\right)+\underset{B\in P\left(X\right)}{\sup }\min \left(N_x^{\theta S}\left(B\right),N_y^{\theta S}\left(B^c\right)\right)\right)+T_S^{\theta }\left({\left\{y\right\}}^c\right)\\ \le \underset{y\ne x}{\inf }\left(1,1+\underset{B\in P\left(X\right)}{\sup }\min \left(N_x^{\theta S}\left(B\right),N_y^{\theta S}\left(B^c\right)\right)\right)\le \underset{y\ne x}{\inf }\underset{B\cap C\ne \varnothing }{\sup }\min \left(N_x^{\theta S}\left(B\right),N_y^{\theta S}\left(C\right)\right)+1\\ =\left[T_2^{S\theta }\left(X,T\right)\right]+1\end{array}$

所以， $\left[T_2^{S\theta }\left(X,T\right)\right]\ge \left[R_S^{\theta }\left(X,T\right)\right]+\left[T_1^{S\theta }\left(X,T\right)\right]-1$ 。

定义7 设 $\Omega$ 是不分明化拓扑空间类，那么称一元模糊谓词 $R_S^{\theta }\in F\left(\Omega \right)$ 为Fuzzifying半 $\theta -N$ 分离的，定义为：

$N_S^{\theta }\left(X,T\right):=\left(\forall A\right)\left(\forall B\right)\left(\left(A\in {\mathcal{F}}_S^{\theta }\right)\wedge \left(B\in {\mathcal{F}}_S^{\theta }\right)\wedge \left(A\cap B=\varnothing \right)\to \left(\exists G\right)\left(\left(G\in T_S^{\theta }\right)\wedge \left(A\subseteq G\right)\wedge \left(C{l}_S^{\theta }\left(G\right)\cap B=\varnothing \right)\right)\right)$

定理11 设 $\left(X,T\right)$ 是Fuzzifying空间，则

$N_S^{\theta }\left(X,T\right)\leftrightarrow \left(\forall A\right)\left(\forall B\right)\left(\left(A\in {\mathcal{F}}_S^{\theta }\right)\wedge \left(B\in T_S^{\theta }\right)\wedge \left(A\subseteq B\right)\to \left(\exists G\right)\left(\left(G\in T_S^{\theta }\right)\wedge \left(A\subseteq G\right)\wedge \left(C{l}_S^{\theta }\left(G\right)\subseteq B\right)\right)\right)$

证明

$\begin{array}{l}\left[N_S^{\theta }\left(X,T\right)\right]\\ =\underset{A\cap B^c=\varnothing }{\inf }\min \left(1,1-{\mathcal{F}}_S^{\theta }\left(A\right)\wedge {\mathcal{F}}_S^{\theta }\left(B^c\right)+\underset{A\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{x\in B^c}{\inf }{N}_x^{\theta S}\left(G^c\right)\right)\right)\\ =\underset{A\subseteq B}{\inf }\min \left(1,1-{\mathcal{F}}_S^{\theta }\left(A\right)\wedge T_S^{\theta }\left(B\right)+\underset{A\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{x\notin B}{\inf }{N}_x^{\theta S}\left(G^c\right)\right)\right)\\ =\underset{A\subseteq B}{\inf }\min \left(1,1-{\mathcal{F}}_S^{\theta }\left(A\right)\wedge T_S^{\theta }\left(B\right)+\underset{A\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{x\notin B}{\inf }\left(1-C{l}_S^{\theta }\left(G\right)\left(x\right)\right)\right)\right)\\ =\left[\left(\forall A\right)\left(\forall B\right)\left(\left(A\in {\mathcal{F}}_S^{\theta }\right)\wedge \left(B\in T_S^{\theta }\right)\wedge \left(A\subseteq B\right)\to \exists G\left(\left(G\in T_S^{\theta }\right)\wedge \left(A\subseteq G\right)\wedge \left(C{l}_S^{\theta }\left(G\right)\subseteq B\right)\right)\right)\right]\end{array}$

定理12 $\models N_S^{\theta }\left(X,T\right)\underset{\dot{}}{\wedge }{T}_1^{S\theta }\left(X,T\right)\to R_S^{\theta }\left(X,T\right)$

证明

$\begin{array}{l}\left[N_S^{\theta }\left(X,T\right)\right]+\left[T_1^{S\theta }\left(X,T\right)\right]\\ =\underset{A\subseteq D^c}{\inf }\min \left(1,1-T_S^{\theta }\left(A^c\right)\wedge T_S^{\theta }\left(D^c\right)+\underset{A\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right)\\ \le \underset{x\notin B}{\inf }\min \left(1,1-\min \left(T_S^{\theta }\left({\left\{x\right\}}^c\right),T_S^{\theta }\left(D^c\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right)\end{array}$

$\begin{array}{l}=\underset{x\notin B}{\inf }\min (1,1-\max \left(1-T_S^{\theta }\left({\left\{x\right\}}^c\right)\right)+\underset{x\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right),\\ 1-T_S^{\theta }\left(D^c\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right))+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right)\\ \le \underset{x\notin B}{\inf }\max (\min \left(1,1-\left(T_S^{\theta }\left({\left\{x\right\}}^c\right)\right)\right)+\underset{x\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right),\\ \min \left(1,1-T_S^{\theta }\left(D^c\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right))\end{array}$

$\begin{array}{l}\le \underset{x\notin B}{\inf }\max \left(1+\underset{x\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right),\min \left(1,1-T_S^{\theta }\left(D^c\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+\underset{z\in X}{\inf }{T}_S^{\theta }\left({\left\{z\right\}}^c\right)\right)\\ \le \underset{x\notin B}{\inf }\max \left(1+\underset{x\subseteq G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right),\min \left(1,1-T_S^{\theta }\left(D^c\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+1\right)\\ \le \underset{x\notin B}{\inf }\left(\min \left(1,1-T_S^{\theta }\left(D^c\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+1\right)\\ \le \underset{x\notin B}{\inf }\left(\min \left(1,1-{\mathcal{F}}_S^{\theta }\left(D\right)\right)+\underset{x\in G}{\sup }\min \left(T_S^{\theta }\left(G\right),\underset{y\in D}{\inf }{N}_y^{\theta S}\left(G^c\right)\right)+1\right)=\left[R_S^{\theta }\left(X,T\right)\right]+1\end{array}$

定义8 设 $\Omega$ 是不分明化拓扑空间类，那么称一元模糊谓词 $R_0^{S\theta }\in F\left(\Omega \right)$ 为Fuzzifying $\theta -R_0$ 分离的，定义为： $R_0^{S\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall y\right)\left(\left(x\ne y\right)\to \left(S{K}_{x,y}^{\theta }\to S{H}_{x,y}^{\theta }\right)\right)$ 。

定义9 设 $\Omega$ 是不分明化拓扑空间类，一元模糊谓词 $R_1^{S\theta }\in F\left(\Omega \right)$ 称为Fuzzifying半 $\theta -R_1$ 分离的，定义为： $R_1^{S\theta }\left(X,T\right):=\left(\forall x\right)\left(\forall y\right)(\left(x\ne y\right)\to \left(S{K}_{x,y}^{\theta }\to S{M}_{x,y}^{\theta }\right))$ 。

定理13 $\models R_1^{S\theta }\left(X,T\right)\to R_0^{S\theta }\left(X,T\right)$

证明

$\begin{array}{l}\left[R_0^{S\theta }\left(X,T\right)\right]\\ =\underset{x\ne y}{\inf }\min \left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\underset{x}{\inf }{\mathcal{F}}_S^{\theta }\left(\left\{x\right\}\right)\right)\\ \ge \underset{x\ne y}{\inf }\min \left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\underset{x\ne y}{\inf }\min \left(\underset{y\notin A}{\sup }{N}_x^{\theta S}\left(A\right),\underset{x\notin B}{\sup }{N}_y^{\theta S}\left(B\right)\right)\right)\\ \ge \underset{x\ne y}{\inf }\left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\underset{x\ne y}{\inf }\underset{A\cap B\ne \varnothing }{\sup }\min \left(N_x^{\theta S}\left(A\right),N_x^{\theta S}\left(B\right)\right)\right)=\left[R_1^{S\theta }\left(X,T\right)\right]\end{array}$

定理14 $\models T_1^{S\theta }\left(X,T\right)\to R_0^{S\theta }\left(X,T\right)$

证明

$\begin{array}{c}\left[R_0^{S\theta }\left(X,T\right)\right]=\underset{x\ne y}{\inf }\min \left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\inf {\mathcal{F}}_S^{\theta }\left(\left\{x\right\}\right)\right)\\ \ge \underset{x\ne y}{\inf }\min \left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\underset{x\ne y}{\inf }\min \left(\underset{y\notin A}{\sup }{N}_x^{\theta S}\left(A\right),\underset{x\notin B}{\sup }{N}_y^{\theta S}\left(B\right)\right)\right)\\ \ge \underset{x\ne y}{\inf }\min \left(\underset{y\notin A}{\sup }{N}_x^{\theta S}\left(A\right),\underset{x\notin B}{\sup }{N}_y^{\theta S}\left(B\right)\right)=\left[T_1^{S\theta }\left(X,T\right)\right]\end{array}$

定理15 $\models R_0^{S\theta }\left(X,T\right)\underset{\dot{}}{\wedge }{T}_0^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)$

证明

$\begin{array}{l}\left[R_0^{S\theta }\left(X,T\right)\right]+\left[T_0^{S\theta }\left(X,T\right)\right]\\ =\underset{x\ne y}{\inf }\min \left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\inf {\mathcal{F}}_S^{\theta }\left(\left\{x\right\}\right)\right)\\ +\underset{x\ne y}{\inf }\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)\le \underset{x\ne y}{\inf }\min \left(1,1+\inf {\mathcal{F}}_S^{\theta }\left(\left\{x\right\}\right)\right)\\ \le 1+\left[T_1^{S\theta }\left(X,T\right)\right]\end{array}$

所以， $\models R_0^{S\theta }\left(X,T\right)\underset{\dot{}}{\wedge }{T}_0^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)$

定理16 $\models T_0^{S\theta }\left(X,T\right)\to \left(R_0^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)\right)$

证明

$\begin{array}{l}\left[T_0^{S\theta }\left(X,T\right)\to \left(R_0^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)\right)\right]\\ =\min \left(1,1-\left[T_0^{S\theta }\left(X,T\right)\right]+\min \left(1,1-\left[R_0^{S\theta }\left(X,T\right)\right]+\left[T_1^{S\theta }\left(X,T\right)\right]\right)\right)\\ =\min \left(1,1-\left(\left[T_0^{S\theta }\left(X,T\right)\right]+\left[R_0^{S\theta }\left(X,T\right)\right]-1\right)+\left[T_1^{S\theta }\left(X,T\right)\right]\right)=1\end{array}$

定理17 $\models R_0^{S\theta }\left(X,T\right)\to \left(T_0^{S\theta }\left(X,T\right)\to T_1^{S\theta }\left(X,T\right)\right)$

证明 证明类似于定理16。

定理18 $\models T_2^{S\theta }\left(X,T\right)\to R_1^{S\theta }\left(X,T\right)$

证明

$\begin{array}{c}\left[R_1^{S\theta }\left(X,T\right)\right]=\underset{x\ne y}{\inf }\left(1,1-\max \left(\left(1-C{l}_S^{\theta }\left\{y\right\}\right)\left(x\right),\left(1-C{l}_S^{\theta }\left\{x\right\}\right)\left(y\right)\right)+\underset{x\ne y}{\inf }\underset{A\cap B\ne \varnothing }{\sup }\min \left(N_x^{\theta S}\left(A\right),N_x^{\theta S}\left(B\right)\right)\right)\\ \ge \underset{x\ne y}{\inf }\underset{A\cap B=\varnothing }{\sup }\min \left(N_{x_1}^{\theta S}\left(A\right),N_{x_2}^{\theta S}\left(B\right)\right)=\left[T_2^{S\theta }\left(X,T\right)\right]\end{array}$

定理19 $\models R_1^{S\theta }\left(X,T\right)\underset{\dot{}}{\wedge }{T}_0^{S\theta }\left(X,T\right)\to T_2^{S\theta }\left(X,T\right)$

证明 证明类似于定理15。

定理20 $\models T_0^{S\theta }\left(X,T\right)\to \left(R_1^{S\theta }\left(X,T\right)\to T_2^{S\theta }\left(X,T\right)\right)$

证明 证明类似于定理16。

定理21 $\models R_1^{S\theta }\left(X,T\right)\to \left(T_0^{S\theta }\left(X,T\right)\to T_2^{S\theta }\left(X,T\right)\right)$

证明 证明类似于定理16。

文章引用

贾文英,王瑞英. Fuzzifying拓扑中的θ-半分离定理
θ-Semiseparation Axioms in Fuzzifying Topology[J]. 理论数学, 2023, 13(08): 2284-2291. https://doi.org/10.12677/PM.2023.138235

参考文献